The Mathematics Ph.D. program at Rutgers includes two qualifying
examinations, a written exam and an oral exam.
The written exam is
taken first and
covers advanced calculus, elementary
topology (metric spaces, compactness, and related topics), and the
material of 501 (real analysis), 503 (complex analysis), and 551
It is offered twice a year, near the beginning of
The syllabus represents a common core of material required
of all Rutgers Ph.D.'s. In particular, the exam is designed with the goal
that a pass on this exam shows a level of
mathematical knowledge and ability
appropriate for teaching the central undergraduate classes in
Each student is required to
take the exam by the beginning of the student's
second year; the program director may allow a student who has
entered with less preparation than the norm to take the exam a
specified number of semesters later.
The exam consists of three separate component exams on different topics: algebra; real analysis and elementary point set
topology; and complex analysis and advanced calculus.
who do not pass the exam do not
retake individual components which they have already passed.
Students who fail this exam may
take it again during the semester following the one in which the exam
was failed. Students who fail on the second attempt or who do not take
the exams on schedule (as determined by the program director) will not
be allowed to continue in the Ph.D. program.
"Free" attempt for entering students Students beginning
graduate work at Rutgers may take
the written qualifying exam
at the beginning of their first semester in the program.
If such a student fails the exam, this will not count
as one of the two attempts that the student is normally allowed;
the student will be allowed two additional attempts at the exam.
The exam is a six hour written exam broken up into three exams, each of which is two hours long. The three
exams cover, respectively, algebra, real analysis and elementary point set topology, and complex analysis and advanced calculus.
Each of these two hour exams consists of two parts. Part I has 3 problems, each of which is mandatory,
and part II has 2 problems,
of which the student is expected to do one. Each
student is expected to submit
solutions to all 3 problems in part I, and 1 out of the 2 problems
in part II.
A complete solution to the January 2011 exam is posted
to serve as a model for students to learn how much justification
and detail one should strive to provide --- these solutions are not worked
out in a timed setting as the solutions to the real exams are, so can
afford to contain more complete arguments; students can get full or close
to full credits when their solutions contain all the key ingredients, with
perhaps less detail than those contained here.
Please note that this exam, like all exams before Summer 2014, followed a
different format. A sample exam in the current format, culled from problems of earlier exams, can be found here
in pdf format.